This is a limitation on the decomposition object: It cannot be saved into .mat file. The background for this is that some types of decompositions are stored in an internal format defined by a library we use: We can't practically save these to a file, and if we tried, there would likely be problems when loading such a mat-file on a different computer.
As a workaround, you could instead store the factors of an LU decomposition of each matrix, and then use this matrix to solve the linear system. Unfortunately, this is less elegant than using the decomposition object.
